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Geometry Problem 393: Triangle, Orthocenter, Circumcircles, Congruence, Collinearity

The figure shows a triangle ABC with the orthocenter D (point where altitudes intersect). Circles 1, 2 (center E), 3 (center F), 4 (center G), and 5 are the circumcircles of triangles ABC, ADB, BDC, ADC, and EFG, respectively. Circle 6 is tangent to circles 2, 3, and 4 at points H, M, and N, respectively. Prove that: (1) Circles 1,2,3,4,5 are congruent. (2) D is the center of circle 5. (3) D is the center of circle 6. (4) Points H,B,M are collinear, H,A,N are collinear, and M,C,N are collinear. (5) Points D,E,H are collinear, D,F,M are collinear, and D,G,N are collinear.

Triangle, Orthocenter, Circumcircles

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